# Find Inverse Of Square Root Functions

Examples, with detailed solutions, on how to find the inverse of of square root functions as well as their domain and range.

 Example 1: Find the inverse function, its domain and range, of the function given by f(x) = √(x - 1) Solution to example 1: Note that the given function is a square root function with domain [1 , + ∞) and range [0, +∞). We first write the given function as an equation as follows y = √(x - 1) Square both sides of the above equation and simplify y 2 = (√(x - 1)) 2 y 2 = x - 1 Solve for x x = y 2 + 1 Change x into y and y into x to obtain the inverse function. f -1(x) = y = x 2 + 1 The domain and range of the inverse function are respectively the range and domain of the given function f. Hence domain and range of f -1 are given by: domain: [0,+ ∞) range: [1 , + ∞) Example 2: Find the inverse, its domain and range, of the function given by f(x) = √(x + 3) - 5 Solution to example 2: Let us first find the domain and range of the given function. Domain of f: (x + 3) ≥ 0 which gives x ≥ - 3 and in interval form [- 3 , + ∞) Range of f: [- 5 , +∞) Write f as an equation. y = √(x + 3) - 5 which gives √(x + 3) = 5 + y Square both sides of the above equation and simplify. (√(x + 3)) 2 = (5 + y) 2 (x + 3) = (5 + y) 2 Solve for x. x = (5 + y) 2 - 3 Interchange x and y to obtain the inverse function f -1(x) = y = (5 + x) 2 - 3 The domain and range of the inverse function are respectively the range and domain of the given function f. Hence domain and range of f -1 are given by: domain: [- 5,+ ∞) range: [- 3 , + ∞) Example 3: Find the inverse, its domain and range, of the function given by f(x) = - √(x 2 -1) ; x ≤ -1 Solution to example 3: Function f given by the formula above is an even function and therefore not a one to one if the domain is the set R. However the domain in our case is given by x ≤ - 1 which makes the given function a one to one and therefore has inverse. Domain of f: (- ∞ , - 1] , given Range: For x in the domain (- ∞ , - 1] , the range of x 2 - 1 is given by [0,+∞), which gives a range of f(x) = - √(x 2 -1) in the interval (- ∞ , 0]. Write f as an equation, square both sides and solve for x, and find the inverse. y = - √(x 2 -1) y 2 = (- √(x 2 -1)) 2 y 2 = x 2 -1 x 2 = y 2 + 1 x = ~+mn~√(y 2 + 1) We now apply the domain of f given by x ≤ -1 to select one of the two solutions above. Hence x = - √(y 2 + 1) Change x into y and y into x to obtain the inverse function. f-1(x) = y = - √(x 2 + 1) The domain and range of f -1 are respectively given by the range and domain of f found above Domain of f -1 is given by: [0 , + ∞) and its range is given by: (- ∞ , -1] Exercises: Find the inverse, its domain and range, of the functions given below 1. f(x) = -2 √(x + 2) - 6 2. g(x) = 2 √(x 2 - 4) + 4 ; x ≥ 2 Answers to above exercises: 1. f -1(x) = (1/4)(x + 6) 2 - 2 ; domain: (-∞ , - 6] Range: [- 2 ; ∞) 2. g -1(x) = √((y - 4) 2 / 4 + 4) ; domain: [4 , +∞) Range: [2 , +∞) More links and references related to the inverse functions. Find the Inverse of a Square Root Function Find the Inverse Functions - Calculator Applications and Use of the Inverse Functions Find the Inverse Function - Questions Find the Inverse Function (1) - Tutorial. Definition of the Inverse Function - Interactive Tutorial Find Inverse Of Cube Root Functions. Find Inverse Of Square Root Functions. Find Inverse Of Logarithmic Functions. Find Inverse Of Exponential Functions.